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* 1820 GENERAL PROGRAM LIBRARYSeasonal Adjustment Program for the IBM 1620 10.3. 0 2 4 i : .
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DISCLAIMER
Although each program has been tested by its contributor, no warranty, express or implied, is made by the contributor or COMMON, as to the accuracy and functioning of the program and related program material, nor shall the fact of distribution constitute any such warranty, and no responsibility is assumed by the contributor or COMMON, in connection therewith.
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COMMON USERS GROUP PROGRAM REVIEW AND EVALUATION (fill out in typewriter, ink or pencil)
Date
o Program No. _ _ _ _ _ _ _ _ _
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Program Name: _________________________________________________________ __
1. Does the abstract adequately describe what the program is and what Yes _ _ No it does?
Comment:-_______________________________ _
2. Does the program do what the abstract says? Yes _ _ No Comment:-________________________________________ __
3. Is the description clear, understandable, and adequate? Yes _ _ No Comment. ____________________________________________ __
4. Are the Operating Instructions understandable and in sufficient detail? Yes No Comm ent:....-________________ ~--~-_:___--~__=_~:_____=~~__:_=__-
Are the Sense Switch options adequately described (if applicable)?
Are the mnemonic labels identified or sufficiently understandable?
Comment
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5. Does the source program compile satisfactorily (if applicable)?
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06. Does the object program run satisfactorily?
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7. Number of test cases run . Are any restrictions as to data, size, range, etc. covered adequately in description?
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8. Does the Program meet the minimal standards of COMMON?
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10. Please list on the back any suggestions to improve the usefulness of the program.
These will be passed onto the author for his consideration.
Please return to:
Mr. Richard L. Pratt
Your Name Company
Data Corporation Address
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Dayton, Ohio 45432 Users Group Code
OTHIS REVIEW FORM IS PART OF THE COMMON ORGANIZATION'S -P-R-O-G-RA-M-R-E-V-IE-W-AND EVALUATION PROCEDURE. NONMEMBERS ARE CORDIALLY INVITED TO PARTICIPATE IN THLS EVALUATION.
11/1/65
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Seasonal Adjustment Program for the IBM 1620
Author: Mr. Sinclair W. Groman Budget Department
United States Borax & Chemical Corp.
3075 Wilshire Bouleva.rd Los Ange les, California
Modifications or revisions to this program:, as they occur, will be announced in the appropriate Catalog of Programs for IBM Data Processing Systems. When such an announce- ment occurs, users should order a complete new program from the Program Information Department.
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DECK KEY
1. 60K Object Deck
2. Example Problem Input Deck 3. Example Problem Output Deck
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4. Source Deck*
This will be forwarded only when specifically reques~ed..'
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I(,ZO USERS Group 1 .. lbrary
ProgrAm AblltrAct
Title
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subroutine .. tAte In Title) SEAJONAL AD JuJ7mCIV",a""t(£/JI1)
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_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - -_ _ _ _ _ _ SubJect CIIl""ificatlon~
Author: Or«anh;Atlon:S
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.t. /lallA X AM, C H£tr'lC.'( c ..Direct; Inq"lre .. to: /
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,..""OV/AliJ ", v Ert-1!je 1..£_1' J"Q,/AgU.AUb «IU'IQ S'I1(1lJtHIA!9 (/utAt1VE ...,StoraRe Requlr.ement .. :_-4C1ijL..1,~,~ • ...l,L_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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Equipment Specification .. :
Memory ZOK 40K 60K / ' K Automatic Divide: Yes / N o Indirect Addr;;;lng: Yes
No
,...-Other Special Features Required l...-X J - ~
Additional Remarks (Include at author's dlscretlon: Languagei Flxed/Floati Relocat- number of times run successfully: Pro-
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'R~~"""''''''''''' __ U.S.BORAX
July 17. 1963
The United States Borax & Chemical Corporation hereby grants permission to International Business Machines Corporation. and in particular to that specific section or group thereof known as "I.B.M. Users Group" to utilize.
apply. experiment with. publish and distribute that particular program developed by Mr. Sinclair W. Groman. Budget Department. United States Borax
& Chemical Corporation. known as "Seasonal Adjustment Program for the I.B.M • 60 K 1620 Computer"; the aforesaid permission is not intended to transfer any rights of ownership our Corporation may have in and to said program nor to preclude the unrestricted use. and independent dist~ibution use and publication of said program. by United States Borax & Chemical Corporation in whatever manner this Corporation. in its sole discretion. may deem advisable.
Practical use and application of this program together with experimental tests utilizing such program have indicated the validity and beneficial usage of such seasonal adjustment program but. in voluntarily offering the aforesaid program to International Business Machines Corporation. the United States Borax & Chemical Corporation makes no representations or warranties. express or implied by law. respecting the results or effects of the use of this system by I.B.M. or any other party. All risks resulting from or in any way connected with the use thereof are expressly assumed by the user. International Business Machines Corporation accepts this sub- mission subject to this disclaimer and agrees not to hold United States Borax & Chemical Corporation liable for the results or effects of any usage of this program and. moreover. agrees to deliver copy of this disclaimer to all distributees of I.B.M. of this program.
Mr. R. C. Dosta Assistant Treasurer
UNITED STATES BORAX & CHEMICAL CORPORATION. S075 WILSHIRE BOULEVARD. LOS ANGELES 5, CALIFORNIA. 3B1-5311 MAIL ADDRESS: P. O. BOX 75128, SANFORO STATION, LOS ANGELES!S, CAL-IFORNIA
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Thi. sea.onal adju.tment high-speed computer program i8 the outgrowth of a .are inten.ive economic, bu.in •••• and .ale. foreca.ting effort undertaken by the Budget Department at U.S. Borax. It was designed primarily to meet the foreca.ting requirements of U.S. Borax, particularly tho.e of the Budget Department, but its range of application. is quite general.
A conscious effort was made to construct a simplified rather than a sophisti- cated . . thematical model, the comp~tational procedure being ba.ed, in.ofar a. po •• ible, on the method used in past year. to .ea.onally adjust the Federal .... rve Board'. series of industrial production indexe.. Since seasonal variations in economic time .eri~s are in fact .ociological phenomena which tead to exhibit approximate patterns a. opposed to precisely defined mathe- matical ab.tractions, the empha.i. upon simplification i. thought to be ju.tified. Of course, more refined seasonal adju.tment program. are available frca other sources, .but in this instance substantial econOBies of computer operation were achieved by avoiding the tendency to over sophisticate the computational procedure.
Perhaps one of the major advantages of this seasonal adjustment program is the amount of control which the analyst responsible for the forecasting function exercises over the manner in which the actual seasonal adjustment occurs. The knowledge, skill, experience, and judgement which the analyst po •• es.e. about any given economic time series may all be incorporated in the seasonal adjustment procedure. Considerable preliminary snalysis mu.t be performed, and provision i. made for manual intervention should it become nece •• ary; it follow. that the analy.t must pos.ess a working knowledge of bu.ine •• , economic, and .ale. foreca.ting technique.. The computer, in other word., serve. as a high-.peed calculating machine capable of communicating with the analy.t, but the burden of decision making remalns with the analyst.
rurthermore, the seasonally adju.ted output deta i. not infallable, and the analyst .hould not he.itate to modify the results If, in hi. judgement, .uch modification i. in order. For the reasons enumerated above, this computer proar .. will probably be mo.t u.eful to those analysts working directly in the foreca.ting function and Who are willing to accept .ome reasonable bal- ance a.ona eODtrol, speed, realism, practicality, and cost. Admittedly, however, the prograa bas it. limitations, and those analy.ts wbo prefer a .ore theoretical computational model will probably find that it was Dot designed to adequately .atisfy their particular requirements. In the final analysls, this .easonal adjustment program is a foreca.ting aid, and it is not intended to be a substitute for experience, knowledge, and skill.
In. closing, it i. appropriate that·mention should be made of the many profes- .ional courtesies extended by the Signal Oil aad Gas Company and Computermat Incorporated, both of whose main offices are in Los Angeles, and the Loa Angele. office of IBM.
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REFERENCE
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2.0
3.0
4.0
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1.1 1.2
2.1 2.2 2.3 2.4
2.S 2.6 2.7
3.1 3.2 3.3 3.4 3.S
4.1 4.2 4.3
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11 TABLE OF CONTENTS
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INTRODUCTORY catmrrS AND INITIAL SINSE SWITCHES SITTINGS INTRODUCTORY CQ4M!NTS • • • •
INITIAL S!HSE SWITCH SETTINGS • • • •
GENERAL DESCRIPTION OF THE SEASONAL ADJUS'l'MDT PROGIAM RAW DATA INPUT
TYPE OF RAW DATA INPUT NUMBER OF Y!AllS OF RAW DATA •
NtlmER OF YEARS IN WHICH SEASONAL OUTPUT IlfFOlHAnOM IS AVAlLABLI • • •
RAW DATA FORHAT •
ORDERING OF RAW DATA INPUT SEASONAL ADJUS'l'HENT PROCEDURE • CARD INPUT ROUTINE
PARAMETER ELEH!IIT • • JOB MtlfBER ELIM!HT IDENTIFICAnON ELEMENT RAW DATA ELP.MIHT CARD INPUT SUMMARY COMPUTA nONA!. PROCEDURE •
MOVING AVERAGE AND SEASONAL FACTOR GENlRAnON lTERATIVI PROCESS ROUTINE • • • • • • • • • • • • • •
LEAST SQUARES DISPERSION REDUCTION RounNE
SEASORAL FACTOR PREDICTION AIm SEASONALLY ADJUSTID SALIS CCMPILAnOR ROUTINE • • •
PRINTER OUTPUT ROUTINE
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2 2 3 3 7 7 1S 15 16 17 17 17 21 22 23
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Page One ECONOMIC TIME SERIES
iii SEASONAL ADJUSTMENT PROGRAM FOR 'mE IBM 60l{ 1620 CcttPUTER
REFE~CB
~R 6.0
6.1 6.2 6.3 7.0
1.1 1.2 7.3 8.0 9.0 10.0
10.1 10.2 U.O
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MANUAL INPUT ROUTINE • • • • • • • • • • • • • • • • • PARAMETER OUTPUT HESSAGE NO. 1 AND EXECUTE INSTRUCTION MESSAGE FOLLOWED
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A PAUSE • • • • • •DECISION TO MANUAL~Y INPUT PARAMETERS • • • DECISION NOT TO }.tANUALLY INPUT PARAMETERS SENSE ROUTINE • •
TO MANUALLY ABORT THE SEASONAL ADJUSn1ENT PROCESS • • • TO MANuALLy BYPASS SEASONAL OUTPUT
TO MANUALLY ENTER THE MANUAL INPUT ROUTINE PUNOl OUTPUT ROUTINE • • • • • • • • • TIME REQUIRED TO OPERATE THE PROGRAM FLOW DIAGRAM
FLOW SUMMATION
DETAILED FLOW DIAGRAM • • • FORTRAN II LISTING
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30 30 31 31 32 32 32 32 40 4l 43 44 56
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1.0 INTRODUCTORY CClo1MENTS AND INITIAL SENSE SWITCH SETTINGS 1.1 INTRODUCTORY COMMENTS
1.2
This seasonal adjustMent program is designed for the IBM 60l{ 1620 computer.
The hardware floating point equipment, if available, is not used and con- sequently the program will operate with all 60l{ 1620 computers. It is true, however, that those computers with hardware floating point equipment will be operating et less than optimum efficiency. This shortcoming, of course, may be corrected by recompiling the FORTRAN II deck to be consistent with
the floating point hardware.
The subroutines are included in the object deck and should not, therefore.
be input separately. The object deck has been condensed in size by a core dump. While it is true that some small excess core capacity exists, for all practical purposes it may be assumed that the entire 60l{ capacity is used up by the program.
The computational procedure, although unsophisticated in the aath . . . tical sense, is based on an iterative procedure and should effectively remove seasonal fluctuations from economic time series. This seasonal adjustment program involves the use of a single object deck, and consequently it is possible to seasonally adjust any number of sets of data in consecutive order, without stopping the computer. The program automatically processes one set of data after another, with no manual intervention needed from the operator. Through the use of card input control parameters, a large degree of flexibility is built into the program, enabling it to meet a wide var- iety of seasonal adjusoaent needs.
No attempt has been made to seasonally adjust data at the midpoint of the computing unit (month, bi-monthly period, or quarter). While seasonal factors may be predicted several years into the future, seasonally adjusted data may be computed only through the end of the last year containing a full year's data. In most instances, this is the year immediately preced- ing the current year. Data for the initial half-year and the final half- year will be lost.
In this seasonal adjustment program certain restrictions apply which should be understood prior to seasonally adjusting data. For example, no attempt has been made to compensate for the variation in the number of working or marketing days in the unit. designating time intervals. Also, the linear least squares technique and a st.ple ratio smoothing method are used to calculate the seasonal factors. Finally, at most eleven years of raw data may be proc8saed by the program. These restrictions, however, do not seriously ~pair the usefulness of the program, although they may cr.at.
obstacles which must be overcome by the application of ingenuity, experience.
and judgement.
INITIAL SENSE S~iITCH SETTINGS
Initially. all four sense switches should be set in the off ~osition.
Should any of the sense switches be thrown on while the computer is proc.a- sing a particular time series, as indicated in various sections of this
report. all sense switches will be thrown off prior to processing the next time series. A printer output message will in this instance inform the operator to turn off all the sense switches.
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~.O GENERAL DESCRIPTION OF THE SEASONAL ADJUSTMENT PROGRAM 2.1 UNIT TIME INCREMENTS OF RAW DATA INPUTS
The seasonal adju.tment program is adaptable to the following unit time increments of raw data inputs:
a. Monthly periods b. Bi-monthly periods c. Quarterly periods 2.2 TYPE OF RAW DATA INPUT
Any consecutive numerical data from an economic time series for which a seasonal pattern is suspected and which conforms to the unit time inter·
vels specified in 2.1 ab~ve may be input.
2.3 NUMBER OF YEARS OF RAW DATA
Up to eleven years of raw data may be input. If seasonally adjusted information is desired for years 1 through N, then raw data for the last half of year zero through the first half of year N + I must be input.
In other words, up to ten complete years of raw data may be input, and two half-years of raw data must be input, for a maxtmum of eleven years.
It is recommended that a mintmum of five years of raw data be input for satisfactory results. In the event that the seasonal pattern i. shifting substantially in a non-linear manner, it is recaa.ended that a . . ximum of six to eight years of raw data be input. The reason for this is to minimize the weight of the earlier years in the linear least squares computational procedure.
2.4 NUMBER OF YEARS IN WHICH SEASONAL OUTPUT INFORMATION IS AVAILABLE Input raw data for the last half of year zero and the first half of year N + I, while processed, is lost for the purposes of seasonal output.
Seasonally adjusted data may be computed for years 1 through N, while seasonally predicted data may be computed for years N + 1 through N + 4.
N can be as high as 10, that is, up to ten years of seasonally adjusted data
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be output, while an additional four years of predicted seasonal factors only may be output.2.5 RAW DATA INPUT FORMAT
The program is written in floating point FORTRAN II without the hardware floating point option. and the raw data input word format must be in whole number (fixed point) notation with a maximum of eight digit positions.
Since nIne digits of seasonally adjusted output data are allowed no output format probleas will normally occur. However, some caution shOUld be ex.rcised in this respect, becau.e eight digits of input data can con- ceivebly result in ~re than nine digits of output data UDder certain exceptional cirCcllll8tancea. In the ev~t that the data for a \lnit time interval is .ero. it is rec~ended that the integer "1" be substituted so as to avoid the possibility of fOlWlt errors. If an adjustment is to be made for the variation in the nualler of working or marketing days in each unit, it will be necessary to input raw data as an average daily rate for each unit. Output data, then, will also be on an average daily rate for each unit.
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Page Three The fonaat for raw date input has been li.plified as auch as polsible.
The rea.on for this. of course, il that the progr . . il intended for ecoaoaic and bUlinesl analysts rather than aatheaatici8DI and pro- greamer •.
2.6 ORDERING OF RAW DATA INPUT
The output format constrains the ordering of raw data input as follows:
2.6.1 IF UNITS ARE NONTHS
If the unit time intervals are months. there are 12 units to a year with unit 1 always being the first month of the year, and unit 12 always being the last month of the year. The particular initial month for each year depends upon what type of a year is involved (calendar, fiscal, administrative, federal, agricultural, etc.) The initial monthly raw data input is for the month six months previous to month 1 o~ year I, and the last monthly raw data input is for the month six months after month 12 of year N, where raw data is input for the coaplete years 1 through N.
2.6.2 IF UNITS ARE BI-MONnILY PERIODS
If unit time intervals are bi-monthly periods, there are 6 units to a year with unit I always being the first bi-monthly period of the year and unit 6 always being the last bi-.anthly period of the year. The particular initial bi-monthly period for each y.ar depends upon what type of year is involved (calendar, fiscal, adainistrat1ve,' federal, agricultural, etc.) The initial bi-monthly period raw data input is for the bi-monthly period three bi-monthly periods previous to bi-monthly period 1 of y . . r 1, and the last bi-monthly period raw data input 1s for the bi-monthly period three bi-monthly periods after bi-.onthly period 6 of year N, vhere raw data is input for the complete years 1 through N. . 2.6.3 IF UNITS ARE QUARTERS
If unit time intervals are quarters there are 4 units to a year with unit 1 always bein~ the first quarter of the year and unit 4 always beiDS the l.at .uarter of the year. The particular initial quarter for each year depends upon what type of year is involved (calendar, fiscal, administrative, federal, agricultural, etc.) The initial quarterly raw data input is for the quarter two quarters previous to quarter 1 of year I, and the last quarterly raw data input is for the quarter two quarters after quarter 4 of year N, where raw data is input for the coaplete years 1 through N.
2.7 SEASONAL ADJUSTMENT PROCEDURE
The seasonll adjusbDent procedure can best be illustrated by a brief description of the program routines.
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2.7.1 CARD INPUT ROUTINE
The following types of data are input on cards:
a. Parameters h. Job Number c. Identification d. Raw Data
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2.7.2 MOVING AVERAGE AND SEASONAL FACTOR CENEAATIClN ITERATIVE PROCESS ROUTINE After the raw data has been read into the computer, an M unit moving average is caaputed, ·where M is the number of units into which each year is divided (twelve for months, six for bi-monthly periods, and four for quarters). From the moving average and raw data, approx- mate seasonal factors are calculated for each unit (monthly period, bi-monthly period, or quarterly period), and linear least square vectors are fitted to the approximate seasonal factors. From the resulting least square vectors, theoretical seasonal factors are generated and smoothed. These smoothed seasonal factors. in con- junction with the raw data, are used to compute a new moving average and a revised moving average halfway between the old moving average and the new moving average. If the new moving average deviates too much from the old moving average. the program will iterate by making another pass through the computational procedure. generating new seasonal factors, a new moving average and a revised moving average. This iterative process continues until the new moving average and old moving average are close enough, or until a pre- determined number of passes have been made, when the program will stop computing seasonal factors.
By card input it is possible to control the program so that only one pass will be made through the moving average and seasonal factor generation iterative process routine. In this instance only the initial moving average, initial approximate seasonal factors, initial least square generated seasonal factors, and initial smoothed seasonal factors will be compiled before program control is trans- ferred out of the moving average and seasonal factor generation iterative process routine. This short-cut path is recommended for those instances where low-profit data is being seasonally adjusted, or for those instances where approximate results suffice.
2.7.3 SEASONAL FACTOR PREDICTION AND SEASONALLY ADJUSTED SALES COMPILATION
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The seasonal factor prediction and seasonally adjusted sales com- pilation routine is an optional feature of the seasonal adjustment program. After the seasonal factors have been generated the program will make a conditional branch to one of the following routines:
a. The seasonal factor prediction and seasonally adjusted sales compilation routine (which is described herein)
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The punch output routine (see 2.7.4 below) c.
If the branch is made to the seasonal factor prediction and season- ally adjusted sales compilation routine, seasonal factors are normally predicted for years N + I through N + 4 by the linear least square vectors generated in 2.7.2 above and seasonally adjusted sales are normally compiled for years 1 through N. The compilation of seasonally adjusted sales may be bypassed by parameter control.
2.7.4 PUNa! OUTPUT ROUTINE
After seasonally adjusted sales have been compiled, the following information is normally punched out:
a. Seasonal factors, including predicted factors b. Least square vectors
c. Seasonally adjusted sales d. Input parameters
e. Compiled counters content
Once the output information has been punched, control will be transferred back to the beginning of the program and the next set of data, if available. will be processed.
2.7.5 LEAST SQUARES DISPERSION REDUCTION ROUTINE
The least squares dispersion reduction routine is an optional feature of tbe seasonal adjustment program. After the seasonal factors have been generated and if a card input conditional branch has been made to the least squares dispersion reduction routine (see 2.7.3 above), the program will check the dispersion of the smoothed seasonal factors about each unit least square line. If the dispersion is too great the seasonal factors will be packed halfway to each unit least square line,a new moving average generated, and control transferred to the moving average and seasonal factor generation iterative process routine (see 2.7.2 for further com- putation.
If the dispersion of the smoothed seasonal factors about the least square line continues to be too great and sufficient passes have been made through the least squares dispersion reduction routine, the program will check to see if the operator shall have the option of manually terminating or continuing the seasonal adjustment procedure. If the operator shall have this option, the program will transfer to the manual input routine (see 2.7.7 below).
If the operator shall not have this option, the program will transfer to the moving average and seasonal factor generation
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Page Six iterative process routine (see 2.7.2) for a final iterative com- pilation of the moving average and seasonal factors. Then. control will be transferred to the seasonal factor prediction and seasonally adjusted sales compilation routine (see 2.7.3). If. on the other hand. the dispersion of the seasonal factors about the least square line is small enough. the program will transfer directly to the seasonal factor prediction and seasonally adjusted sales compila- tion routine (2.7.3 above).
2.7.6 SENSE ROUTINE
The senae routine is· an optional feature of the seasonal adjustment program. Entry to the sense routine is always made by throwing on sense switch No. 2 during the operation of the program. Entry is made into the sense r~utine for the following reasons:
a. To manually abo~t the seasonal adjustment process b. To manually bypass seasonal output
c. To manually enter the manual input routine 2.7.7 MANUAL INPUT ROUTINE
The manual input routine is an optional feature of the seasonal adjustment program. Entry into the manual input routine may be made in one of two ways:
a. From the sense routine under certain circumstances (see 2.7.6 above)
b. When the least squares disper.ion reduction routine indicator is on and the dispersion about the unit least square lines remain8too great and sufficient passes through the least squares dispersion reduction routine have been made and the' operator shall have the option of terminating or continuing the seasonal adjustment procedure (see 2.7.5 above)
The manual input routine will be used in those rare circumstances when. for some reason. it is desired to visually observe the card input parameters in order that the operator may make a decision concerning the continuation or discontinuation of the iterative process. When program control is transferred to the manual input routine the input parameters are printed out on the typewriter.
following which a program PAUSE occurs. If sense switch 1 is thrown on the computer will ACCEPT new parameters. After the parameters are correctly inserted from the typewriter. control is transferred to the moving average and seasonal factor generation iterative process routine for further compilation. If sense switch 1 is left in the off position. control is transferred directly to the moving average and seasonal factor generation iterative process routine for one last pass.
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As the program processes data the printer will output various types of messages which provide the following information:
a. Identification of data being processed b. Trace the path through the program
c. Inform the operator of the results of the iterative process d. Inform the operator of the disperSion of moving average points
and smoothed seasonal factors e. Print out parameters. if desired
f. Inform the operator to make a decision to terminate iteration or to continue iteration
g. Inform the operator to turn off sense switches before the next set of data can be processed
In general. the printer output provides a real time basis for the operator to visually observe the degree of seasonal adjus~nt
which occurs and to perait the operator to take a flexible course of action while the program i. operating, should this become necessary. The courses of action (see 2.7.6 and 2.7.7 above) which the operator may tate are the following:
a. Manually abort the seasonal adjustment process b. Bypass seasonal output
c. Print out parameters
d. Manually input parameters (continue to iterate) or ter.inate iteration
3.0 CARD INPUT ROUTINE
The card input routine will he specified in the order that its elements appear in the card input deck and are read into the computer.
3.1 PARAMETER ELEMENT
The card input parameters control the operation of the progr... lending to the program a great degree of flexibility and providing for the satis- faction of a large variety of seaaonal adjustment requirements. There are 21 parameters. identified as KI through K2l, input for each set of raw data. six parameters to a card. Consequently. the par . . . ter element consists of four cards. The least significant digit of each parameter will appear in card positions 10. 20. 30. 40, 50, and 60 respectively, with the subscripted order of par . . . ters increasing consecutively fro.
left to right and from par . . . ter card No. 1 to parameter card No.4.
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KI. Kl is the first year for Which raw data is input. Kl corresponds to year zero. the last half only of which contains raw data. Kl contains four digit positions and will be punched in card positions 7 - 10 of parameter card No.1.
Example: 1955
K2' K2 is the total number of years of raw data input. including the . last half of year zero, and the first half of year N + 1. K2 is equal to the number of complete years for Which raw data is input plus one.
One is added to the number of complete years for Which raw data is input in order to compensate for the last half of year zero and the first half of year N + I, which also contain raw data. K2 will be
K3.
K4.
KS'
limited to two digit positions, with a maximum magnitude of eleven and a suggested mintmum magnitude of five. K2 will be punched in card positions 19-20 of paraaeter card No.1.
Example: 08
K3 is the number of units in a year. When monthly raw data is input.
K3 = 12; when bi-monthly raw data is input. K3 • 6; When quarterly raw data is input, K3 = 4. K3 is limited to two digit positions and will be punched in card positions 29-30 of parameter card No.1.
Example: 06
K4
is thespeaking, depending desired.
number of years of predicted seasonal factors. Generally
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K4 will be punched in card position 40 of parameter card No.1.
Example: 4
K5 specifies the product of 1000 times the maximum allowable ab- solute percentage moving average deviation for each pair of points.
When the program compares the new moving average with the old moving average. point by point. it calculates the absolute per- centage deviation between the two. If this percentage deviation is greater in absolute magnitude than that specified by KS. counter MPl is incremented by one. K5 is entered into the computer as a whole number and divided by 1000 by the program to obtain the correct perc __ ntage. The program is designed to operate optimally when K5 ~ 5, which, when divided by 1000, means the program screens all moving average deviations in exce.s of one-half of one per cent. However.
K5 is not limited to 5, and it may take on any value 0 ~ K5 ~lOOO.
It is recommended that, for best results, K5 not exceed 20 because when K5 is greater than 20 the program will screen only those moving average deviations in excess of two per cent, that is, counter MPl
~ ()
Page Nine will not be incremented if an absolute deviation of less than two per cent is detected. High values of K5 result in speeding up the seasonal adjustment process; low values of "'5 result in a more reHable seasonal adjustment process. For practical purposes, a balance between the . two must be achieved. and for this reason 5 fo: KS ~ 20 is recOI'lWIIenderi. ~'5
is punched in positions 47 - 50 of parameter card No.1.
Example: 0005
"'-6.
K6 is the maximum number of moving average passes allowed in the moving sverage and seasonal factor generation iterative process routine. Each time the program enters this routine from outside the routine the number of passes it makes through the routine may notexceed~. It may make fewer passes than
Ku,
however, 1f the sea- sonal adjustment process meets the specifications of parameters KS' KlO and K12. As a practical matter, K6 should not exceed 10, but it may be anywhere in the range 0 ~ K6 !Z 1000. At the conclusion of each pass counter NHl is incremented by one. It should be noted that program control may enter the moving average and seasonal factor generation iterative process routine more than once, and each time entry is so made,Kv
specifies the maxtmum number of al- lowable passes while program control is in this routine. K6 will be punched in card positions 57 - 60 of parameter card No.1.Example: 0004
K7• In t?e event that control is transferred from the least squares dispersion reduction routine back to the moving average and seasonal factor generation iterative process routine,
Ko
takes on the value of K7. A substantial amount of seasonal adjustment will have occurred by the time the transfer of control back to the seasonal factor gen- eration iterative process routine is made. and a reduction in the value of ~ may be appropriate. This facility is built into the program in order to take ~dvantage of these potential savings. While it is possible for K7 to be larger than l~. for practical purposes this possibility may be ignored. If a reduction inKo
is desired, 1<7 .c... K6. while i f no reduction in K6 is desired, K7 - K6. K7 will be punched in card positions 7 - 10 of parameter card No.2.Exmple: 0002
Y8. l~ specifies the product of 1000 times the maximum allowable absol- ute percentage seasonal factor deviation. If the least squares dispersion reduction routine is being operated, the program will compare, for each unit, every smoothed seasonal factor with the corresponding least square generated seasonal factor for years 1 through N. If this percentage deviation is greater in absolute magnitude than that specified by Kg, counter NQl is incremented by 1.
I~ is entered into the computer as a whole number and divided by 1000 by the program to obtain the correct percentage. The program i8 designed to operate opttmally when Kg ~;;{ 5, which. When divided by 1000, mean.
o
I I i
I
I
I
~'
•
o
'age Ten
the program screens all seasonal factor deviations in excess of one- half of one percent. novever, KS is not limited to 5, and it l&.ty
take on any -alae 0
"Y. e :
1000. I t is recOlllBended that, for oest results. Ke not exceed 20 because when KS is greater than 20 the program will screen only those a . . 8Onal ractor deviatioILs in «!xcess of two percent. that is, counter NQl viII not be incremented if an absolute deviation of less than two percent is detected. ,ligh values ofKa
result in speeding up the s . . sonal adjustment procesa. lov values of K8 result in a more reliable seasonal adjuatment process.For practical purposes. a balancebetveen the two must be achieved, and for this reason 5 ~ KS ~ 20 is recOlllllended. Ka vill be punched in card positions 17 - 20 of par_eter card tlo. 2.
•. Ex.ple: 0005
J9. J9
.pecifie. the aax~ nuaber of allowable passe. through the leaat square. diaper.ion reiuction routine. It is reca.Dended that the rana_ 0 ~ Kg " 5 apply, althoup K9 .. , be as high as 100. Each tt.e a pa.. is made, NHl i. 1Dcr_ted by one. When no more passes are avdlable, that h, when NHl .~, control will be tran.ferred to the seasonal factor prediction and seasonally adjusted sales ca.pilation routine after the moving average and seasonal factor generation iterative process routine is operated on for one last time.~ will b_ punched in card positions 28 - 30 of parameter card No.2.
Ex.-ple: 003
KIO· KIO apecifie. the maxtmu. number of moving average absolute percentage devi.tiona which . . y exceed that specified by KS and atill be accept- able. KIO, in other word., indicate. the max~ value which counter MPI can b. and the aeasonal adju.~t process remain under control.
Every ttme an exces.ive moving average absolute percentage deviation 18 detected, COURter MPI h incr-.nted by one. After an aoving average . . . nitude deviations are ex..ined, counter MPI is tested with KIO· If the content. of MPl exceed KIO' the test fails, that is, too aaRy extr . . . points have been detected. If the test fails and K6 exceeds the contents of counter NHl (meaning at least one more pas8 1s available through the moving average and seasonal factor generation iterative process routine) another pass viii be made through this routine. If the test i. OK, K12 is tested. The value of KlO will depend upon whether data for month., bi-monthly periods, or quarters is being seasonally adjusted. The suggested formula for deteraining KlO appears below:
a. Monthly data:
KIO • 2(K2 -I), where K2 • number of years of raw data, including last half of year zero and first half of year N + 1.
b. Bi-monthly and quarterly data:
KlO • K2 - I, where K2 i8 defined above.
o n
Page Eleven In both instances a fev extr . . . points are reca.mended, an average of two per year when dealing with monthly data and an average of one per year when dealins with bi-monthly or quarterly data. However, if more accuracy is deSired, KlO ~ 0 might be considered. For monthly data the range 0 ~ KIO ~ 120 applies; for bi-monthly data
the range 0 ~ rIO
s
60 applies; for quarterly data the range 0 ~KIO~ 40 applies. It is probably true that anything greater than the suggested values of KIO in a and b above are not very practical. KlO will be punched in card positions 37 - 40 of parameter card No.2.Example: 0012
If the test sumaation results are printed on the typewriter, the most recent result of this ,test will be indicated •
Kll. Kli s~cifies the max~ number of aeasonal factor absolute per- centage deviation. which . . y exceed that specified by Kg and still be acceptable. Kl l ' in other word., indicates the aax~ value which counter NQl can be and the seasonal adjus~nt process ra.ain under control. EYery ttme an excessive seasonal factor abaolute per- centage deviation is detected, COURter NQl is incr.-.nted by one.
Aftar all seasonal factor .agnitude deviations are ~ned. counter 'NQl is tested with Kil' If the contents of NQl exceed Kll, the test fails, that is, too many extr ... points have been detected. If the test fails and Kg exceeds the contenta of NHl (meanina at least ODe more pass is available through the l . . at aquares dispersion reduction routine) another pass will be . . de throuch both the aoving average and seasonal factor generation iterative pror.ess routine and the least square diapelsion reduction routine. If the test is OK, Kl3 viii be teated. If the test fall. and no more paases are available, KlS will be tested after the te.t ... tion results are printed.
The value of Kll will depend upon vbether data for months, bi-monthly periods, or quarters is being seasonally adjusted. The suggested method for determining Kil appears below:
a. Monthly data:
Kll • .2 (K2 - I), where minimua Kll • 1 b. Bi-monthly data and quarterly data:
Kil Q .2 (K2 - I), where aintmu. Kll • 1
In each instance at least one extr ... point ia reco.aended. However, if more accuracy is desired, Kll • 0 might be considered. In each instance the ranae 0 ~ Kll , 10 applies. It is probably true th.t anything greater than the suggested value of Kll ia not very practical.
Kil viII be punched in card poaitions 49-50 of parameter card No •. 2.
Ex . . ple: 02
•
-.
,
K12 •
K13'
Page TWelve If the test sumPation results are printed on the typewriter, the Qost recent result of this test will be indicated. K12 specifietJ the max- t.um allowable sum of the absolute percentages of the moving average deviations. When the progr_ detemines the percent deviation hetween the old moving average and the new moving average, point by point, it squares each deviation to eliminate tne sign, multiplies by 1,000,000 to elbainate the decbDal point, and then takes the square root. If an absolute deviation is in the neighborhood of .005 (one-half of one percent), the final result of the above calculation will be in the neighborhood of 5. If, on the other hand, an absolute deviation is In the neighborhood of .02 (two percent), the final result of the above calculation will be in the neighborhood of 20. Kl2 is used to test the sum of the absolute deviations which have been calculateJ in the manner indicated above. The f~11owing formula for determining ~12 is su;sested;
a. Monthly data:
K12 • l2K5 (K2 - 1) b. Bimonthly data:
K12 • 6K5 (l2 - 1) c. Quarterly data:
K12 • 4K5 (K2 - 1)
While the range 0 ~ K5 ~ LOO applies, it is strongly recOll1illended that K5 not be allowed to exceed 20, because values in excess of 20 mean that average absolute deviations in excess of 2t will be accepted.
The sum of the moving average deviations are accumulateu in counter PMl. After the deviations are accumulatec, counter F1:l is teste~
with K12' I f the contents of K12 exceed the counter the test is OK;
if not. the test fails. If the test fails and K6 exceeds the contents of counter NUl (meaning at least one more pass is available through the moving average and seasonal factor generation iterative process routine) another pass will be made through this routine. If the test is OK, or if no more passes are available, Kl8 is tested. K12 will be punched in card positions 57 - 60 of par8CIeter card t·:o. 2.
Example: 0600
Kl3 specifies the maximum allowable sum of the absolute percentages of the seasonal factor deviations. When the progr . . determines the percent deviation between the ..aothed seasonal factors and the corresponding least square generated seasonal factors it squares each deviation to eliminate the sign, multiplies by 1.000,000 to eliminate the decimal point, and then takes the square root. If an absolute deviation is in the neighborhood of .005 (one-half of one percent) the final result of the above calculation will be in the neighborhood of 5. If, on the other hand, an absolute deviation is in the neighborhood of .02 (two percent), the final result of the above calculation will be in the neighborhood of 20. K13 is used to test the sum of the absolute deviations which haile been calculated in the manner indicated above. The following formula for determin- ing K13 i8 suggested:
~ rJ
a. Honthly data:
K13 '" K8 (K2 - 1),
h. Bi-monthly d(lta and quarterly data:
113 • K8 (r2 - I),
Page Thirteen
While the range 0 ~KR ~ 100 applies, it :f.S stronsly recOIIIDendcd that Ka not be allowed to exceed 1(', becau Je values in excess of 10 mean that average absolute deviations in excess of 1% be accepted. The
Slim of the seasonal factor deviations are accumulated in counter FNI.
After the deviations are accumulated, counter FlU is tested with K13.
If the contents of K13 exceed the counter the test ia OK, if not, ~be
test fails. If the ~est fails and ~ exceeds the contents of NHl (meaning at least one more pass is available through the least squares dispersion reduction routine). another pass will be .ade through both the moving average and seasonal factor generation iterative proce88 routine and the least square dispersion reduction routine. If the test is OK the program will test K19' If the test fails and no .ore passes are available (ID-1l exceeds Kg), KlS will be tested after the test summation results are printed. K13 will be punched in card positions 8 - 10 of parameter card No.3.
Example: 050
K14' K14' the seasonal factor prediction bypas8 parameter, may assume three values, 0, I, and 2, but only in the manner specified below with respect to K17 • Violations of thi8 speCification may re8ult in incorrectly punched cards. When K14 • 0 and Kl7 • 0, least square vectors are fitted to the seasonal factors and the progr . . processes data in a normal . . nneI'. When Kl4 • 1 and K17 • 2 least square vectors are fitted to the sea80nal factors only once. after Whicb program control bypasses the seasonal factor prediction and .easonally adjuated sales compilation routine, transferring control directly to the punch output routine. If K14 • 2 and K17 - 0, least square vectors will not be fitted to the seasonal factors and the seasonal factor prediction and seasonally adjusted sal.s compilation routine is bypassed, program control being transferred directly to the punch output routine. K14 must be used only in conjunction with K17 in . the manner specified. K14 will be punched in card position 20 of
parameter card No.3.
Example:
K15. Kl5 is the manual input switch. After the test summation results have been printed, the program will test K15. If KlS - 0, the program will cycle through the moving average and sessonal factor generation iterative process routine for the last time. If K15 • 1. the program will enter the sense routine to check for any of the sense switches being turned on. If none of the sense switches are turned on. or if
o .'
•
o
Page Fourteen only sense switch No. 2 is turned on, the program will enter the manual input routine. K15 will be punched in card position 30 of parameter card No.3.
Example:
K16. K16 is a spare parameter which should always be zero. K16 will be punched in card position 40 of parameter card No.3.
Exaaple: 0
K17' K17 is the least equares fit bypass par ... ter, and it may assume
KlB'
K19'
three values, 0, I, and 2. When K17
=
0 or 2, the progr . . will pro- cess data in the manner specified by the description of parameter Kl4 above. If K17 • I, seasonal factors are computed once without using least square vectors ~nd then control is transferred directly to the punch output routine without predicting .easonal factors and seasonally adjusting .ales. When tl7 - I, Kl4 should be zero, since Kl4 will never be tested in this instance. K17 is punched in card position 50 of para.eter card No.3.Example: 0
KlB is the dispersion s.ooth switch, and it may assume three values, 0, I, and 2. When KlB • O. the program will bypass the least squares dispersion reduction routine and will directly enter the seasonal factor prediction and seasonally adjusted sales c~ilation routine.
When KlB • 1. the program will enter the least squares dispersion reduction routine. When KIB - 2, the program will bypass the least squares dispersion reduction routine. in addition, the program will also bypa.s caaputing seasonally adjusting sales and it will permit only tha seasonal factors to be punched out. When K14 is 1 and K17 is 2, or when K14 1s 2 and K17 is O. K18 is always 2, because in the.e instances only seasonal factors are to be punched. KlB will be punched in card position 60 of parameter card No.3.
~le: 1
Kl9 i. the parameter which tests the sum of the combined moving average deviations and the .... onal factor deviation. when the disper.ion 8mOOth .witch KIB • 1. The suggested fo~ula [or
ce-
termining Kl9 is as follows: K19· K12 + KIJ'
Counter NP4. which is the sum of the mean FNI .. pf" ('lUll t~e mean PMl • PM. is tested with K19' If Kl9 exceecs NP4, the te~t is Of- and the program viII transfer control to the seasonal factor pre- diction and the seasonally adjusted sales cOIIIpi lation routine.
If the content'! of counter NP4 exceed K19' the test fails. I f the test fails and Kg exceeds the contents o! NIH (meaning at leaH one more pass 15 available through the least squares
o
3.2
3.3
1
o
Page Fift . . n dispersion reduction routine), another pass will be made throUlh both the moving average and seasonal factor generation iterative process routine and the least square disper.ion reduction routine. If the test fails and no more passes are available (NMI exceeds Kg), K15 will be tested after the test summation results are printed. K19 will be punched in card p08itions 7 - 10 of par . . . ter card No.4.
Example: 0010
K20. K20 is the print suppress indicator, and it may be either zero or ODe.
If K20 is zero, all printer output . . ssages are printed. If K20 is one, the Job number printer message, both identification printer messages, and the end of data printer . . ssage are bypassed. K20 will be punched in' card position 20 of parameter card No.4.
Ex.-p1e: 0
K2l. K21 is a .pare parameter which should alway. be zero. K2l will be punched in card position 30 of par ... ter card No.4.
Exaaple: 0 JOB NUMBER ELEMENT
The job number identifies a particular data batch or run, and it . . y be used to sort a deck of output cards should they be dropped or aingled with another output deck. The Job number will be punched twice on a single card, together with the initial card counter or LO number of one. The job nusher is composed of three digits, and will be punched in card po.itions 8 - 10 and again in card positions 63 - 65. The card counter or LO number, which is alway. one in this instance, will be punched in card po.ition 71.
Example: 8 - 10 001 IDENTIFICATION ELEMENT
63 - 65 001
71 1
The purpose of the identification element is to identify the . . a80Dally adju8ted data which will be punched on cards. The identification el ... nt is coaposed of two cards, each of which contain. up to 39 alpba.erical characters and a card counter or LO uu.ber.
a. Identification card No. 1 will contain alphaaerical information punched in card positions 1 - 39, and card counter or LO nuaber two punched in card position 71. This identification information may be anything deeaed to be of value in identifying the data.
Ex_pie: 20 MTP PACK BOR HO LB SLS 2
b. Identification card No. 2 will contain alphamerical information punched in card positions 1 - 39, and card counter or LO nuaber three punched in card po.ition 71. This identification inf~raatiOD may be anything de . . . d to be of value in identifying the data.
Exaple: SEAS ADJ FR~ 1952 3
• ..
Pa~e Sixteen
3.4 RAW DATA ELEMENT
The raw data element will have three separate formats, depending upon whether monthly, bimonthly, or quarterly data is being input. Under no circumstances will more than 132 months or 66 titlonthly periods or 44 quarters of raw data be input, that i~, the program will process at most eleven years of raw data. In addition, it'i9 recommended that a minimum of five y .. ars of raw data be input for reliable results. P~w
data must be input in consecutively ordered cards, toe initial card containing the oldest data and the last card containing the most recent data. Each input data word must be in whole number (fixed point) notation with a maximum of eight digit, positions. Data wi 11 be punched starting with the last half of year zero and ending with the first half of year N + 1.
In the ovent that the data for. unit time interval is zero, it is recommended that the integer "1" be substituted so as to avoid tile pos- sibility of fo~t errors (see 4.1.2).
3.4.1 MONtHLY DATA
Each monthly data card must contain exactly six consecutive months of data, with the least significant ,!igit of the data of thf!
initial month being punched in card positiun 10 and the leaat significant digit of· the data of the 1Il0st recent month being punched in card position 60. The least significant digits of the data of the intervening consecutive months will be punched in card positions 20, 30, 40, and SO respectively, beginning with the data of the second oldest month and ending with the data of the second most recent month. In this manner cata cards will be consecutively ordered. and the data on each card will also be consecutively orderev, resulting in a consecutively array of raw data from the initial month to the most recent month.
Example:
3 - 10 13 ~ 20 00004126 00005283
23 - 30 00006718 3.4.2 B~10NT.ILY PE~IOD ~tA
33- 40 0000887/.
43 - 50 00006112
53 - 60 00003009
Each bimonthly data card must contain exactly three consecutive bimonthly periods of data, followed by three consecutive zeros, The least significant digit of the data of ttle initial :-imollthly period vi 11 be panched in cart) position 10, the least sign1 ficanc digit of the data of the middle bimontbly per1~jtl w1l1 h1 pt;nched in card position 20, and the least significant digit of the data of the most recent bimonthly period wi 11 be p',nC'hed in cara t'0s~
it1:'" 30. Zeros wi1l be punched in card pOSitions 40, 50, and 60.
Example:
3 - 10 00002215
()
13 - 20 00003342
23 - 30 00006952
40
o
50 o 60o
4.0
(l
3.5
Page Seventeen 3.4.3 QUARTERLY PERIOD DATA
Each quarte~ly data card must contain ,exactly two consecutive quarters of data. followed by four consecutive zeroes. Tbe least significant digit of the data of the initial quarter will be punched in card pOSition 10, and the l.ast significant digit of the data of the moat recent quarter will be punched in card position 20. Zeroes will be punched in card.positiona 30, 40, 50, and 60.
3 - 10
Example: 00001513 13 - 20 00006298
30
o 40
o
50o
60 oCARD INPUT SUMMARY
Cards will b. input in the following order:
Card 1. Par ... ter card No. 1 (KI -.
R6)
Card 2. Par ... ter card No. 2 (K7 - K12) Card 3. Parameter card No. 3 (K13 - KIS) Card 4. Parameter card No.4' (K19 - K2l ) Card S. Job maber card
Card 6. Identification card No.
Card 7. Identification card No.2
Card 8 thrGugh laat card. Raw data cards in conaecutive order, beginning with raw data for last half of year zero and ending with raw data for first half of year N + 1.
<n!PUTATIONAL PROCEDURE
4.1 MOVING AVERAGE AND SEASONAL FACTOR GENERATION ITERATIVE PROCESS ROUTINE 4.1.1 MOVING AVERAGE COMPUtATION
Let j be any unit in the time interval. After the input data hes been read into the computer, for each unit j the program computes a moving averase Aj from the raw data Dj based on a l2-unit period if monthly data is used or a 6-unit period if bi-monthly data is used or a 4-unlt period i f quarterly data is used. TIle IDOving average 1s computed by the following formula:
A
=
D(j _ Ml2) + ••• + Dj + ••• + D«j + Ml2) _ 1) MWhere: M - number of units in a year and j - the unit number.
Furthermore, if N = K2 - I, where N is the number of years, then the following reiationship holds: 1 ~ j ~ MN. This relationship
()
..
•
.:
o
Page rightoaen hold. true for all coaaputaUons. Note that a vector rati>c!' thau !I
matrix ootaUon 18 used to order increasing valuu of j.
'l'hls .ovtna average strects fr~ the raw (~ata the [tnt estimate o.f Cho.e time .eries fluetationa attributat-le to sea$o:lal causes, because each month, bimonthly pcriorl, or quarter is given equal waight over a one-year interval.
It.l.2 APPI!OXIKA'I! SEASONAL FACTOIS
After tha inlttal .ovina average Aj is cC*lputect , the raw data Dj for the len
"-If
of year .ero and the f1 rat ha 1f of year !i .... 1 is dn,pad f.,. the cOllPtiUng 1DOdel. Hereafter only rav dataD,
fory . .
r.
I thl'Oulh H "ill be relevant. For each unit j, approximate .... anal f.ctora Rj-!1
.re caaputed .. The.e approxi . . te sea.onalA
f.ctor. are ratios
vhlC~
provide an estimate of the variation in an.c ....
te tia. a.rie. whlch . . y be attrib~table to aea80nal causes.POl' very _11 A" fonaat error . . . y occur. In order to avoid very _ l l Aj' aub.titute the integer "I" for zero whenaver zero appears a. the raw data input for a unit time interval.
4.1.3 FOWUJ.,A LINEAR LEAST SQUARES FIT OF APPROXD'.ATE SEASONAL FACTORS T .... r.rtly. j Will refer to unit., i will refer to years, and the . ~ . . td_notation "ttl be used for the purposes of illustration only.
rOI' . . eh unit j Coraula linear least square Unes are fit to the
approa'aate seasonal factors Rj where the Rj for each unit are orQl'ed by 1 frOll year 1 throup y .. r N - K2 • 1. The formula unit 11 . . . r l . . . t square lines are coaputed from the following set of . . . tiona:
i1 j -Na + bit 1'1.1 •
4:i
+ btl Whare:....
ADd:Aftd:
a • constant tel'll
b • 11near coefficient of independent variable i • year nu.ber
J - unit nWlber
N-.axi.ua year nuabar - nuaber of years I 11 j • 12 for IIOnths
I $-j • 6 for bi.athly periods
r ..
J So. "4 for tlUartet'8 1 ! I5
12 • 1Ri,.j • seasonal factor of the i th year for the j th uni t Thare are tvelve, six or four sets of equations depending upon whethar _nthly, biaonthly, or quarterly data, respectbely, is bains 'roe •••• d. From th.se sets of e~uatlons tvelve, six, or four unit lea.t sfluare fonnula. of the following f01'llll are computed:
Ilt.J • aj + bji
o o
page Nineteen Where: Ri,j now refers to formula seasonal factors as opposed to approximate seasonal factors.
Although the vector notation of j is used in the progr.m computa- tional procedure, the matrix notation 1,j is used here for the purpose of illustration only. The final result is the same 1n both instances, excepting that the computer operates faster uslns the vector notation rather than the matrix notation.
4.1.4 GENERATION OF FORtnn.A SEASONAL FACTORS
Once again the matrix notation rather the vector notation t. used for the purpose of illustration only. The formula least square seasonal factors are generated from the following set of equations, where there are as meny equations as there are units j.
Ri,j • 8j + bjt
Since a formula has been computed for each unit it is possible to generate formula seasonal factors for each unit of each year i from year I through year N • K2 - 1.
4.1.5 SMOOTIHNG FORMULA SEASONAL FACTORS
Formula seasonal factors have been generated by unit equationa;
however, for a given year each seasonal factor should average 1.0.
This means, if monthly data is being processed, the sua of the fo~ula seasonal factors for each year should a.Dunt to 12.0; if bi-monthly data is being processed. the annual total should be 6.0;
if quarterly data is being processed, the annual total should be 4.0.
If FK3 = require. sum of the seasonal factors Ri for year t. and ZYR
=
actual sum for year i, then the smoothing factor for year i, SZYR, is found by:SZYR • (FK3 - ZYR)/ZYR
For each year i, the formula seasonal factors Ri . . y be smoothed, if smoothing is necessary, by the following formula:
Ri • Ri + Ri (SZYR)
The computational procedure for smoothing seasonal factors is illustrated below (assuming hi-monthly periods):
